Changes

A binary operation or LOC <math>*\!</math> on <math>X\!</math> is '''commutative''' if and only if <math>x*y = y*x\!</math> for every <math>x, y \in X.\!</math>

A binary operation or LOC <math>*\!</math> on <math>X\!</math> is '''commutative''' if and only if <math>x*y = y*x\!</math> for every <math>x, y \in X.\!</math>

A '''semigroup''' consists of a nonempty set with an associative LOC on it.  On formal occasions, a semigroup is introduced by means a formula like <math>\underline{X} = (X, *),\!</math> read to say that <math>\underline{X}\!</math> is the ordered pair <math>(X, *).\!</math>  This form specifies <math>X\!</math> as the nonempty set and <math>*\!</math> as the associative LOC.  By way of recalling the extra structure, this specification underscores the name of the set <math>X\!</math> to form the name of the semigroup <math>\underline{X}.\!</math>  In contexts where there is only one semigroup being discussed, or where the additional structure is otherwise understood, it is common practice to call the semigroup by the same name as the underlying set.  In contexts where more than one semigroup is formed on the same set, one can use notations like <math>\underline{X}_i = (X, *_i)\!</math> to distinguish them.

A '''semigroup''' consists of a nonempty set with an associative LOC on it.  On formal occasions, a semigroup is introduced by means a formula like <math>\underline{X} = (X, *),\!</math> read to say that <math>\underline{X}\!</math> is the ordered pair <math>(X, *).\!</math>  This form specifies <math>X\!</math> as the nonempty set and <math>*\!</math> as the associative LOC.  By way of recalling the extra structure, this specification underscores the name of the set <math>X\!</math> to form the name of the semigroup <math>\underline{X}.\!</math>  In contexts where there is only one semigroup being discussed, or where the additional structure is otherwise understood, it is common practice to call the semigroup by the same name as the underlying set.  In contexts where more than one semigroup is formed on the same set, indexed notations like <math>\underline{X}_i = (X, *_i)\!</math> may be used to distinguish them.

A '''unit element''' in a semigroup <math>\underline{X} = (X, *)\!</math> is an element <math>e\!</math> in <math>X\!</math> such that <math>x*e = x = e*x\!</math> for all <math>x \in X.\!</math> In other words, a unit element is a two-sided identity element.  If a semigroup <math>\underline{X}\!</math> has a unit element, then it is unique, since if <math>e'\!</math> is also a unit element, then <math>e' = e'*e = e.\!</math>

A "unit element" in a semigroup X = <X, *> is an element e in X such that x*e = x = e*x for all x C X.  In other words, a unit element is a two sided identity element.  If a semigroup X has a unit element, then it is unique, since if e' is also a unit element, then e' = e'*e = e.

A "monoid" is a semigroup with a unit element.  Formally, a monoid X is an ordered triple <X, *, e>, where X is a set, * is an associative LOC on the set X, and e is the unit element in the semigroup <X, *>.

A '''monoid''' is a semigroup with a unit element.  Formally, a monoid <math>\underline{X}\!</math> is an ordered triple <math>(X, *, e),\!</math> where <math>X\!</math> is a set, <math>*\!</math> is an associative LOC on the set <math>X,\!</math> and <math>e\!</math> is the unit element in the semigroup <math>(X, *).\!</math>

An "inverse" of an element x in a monoid X = <X, *, e> is an element y C X such that x*y = e = y*x.  An element that has an inverse in X is said to be "invertible" (relative to * and e).  If x C X has an inverse, then it is unique to x.  To see this, suppose that y' is also an inverse of x.  Then it follows that:

An "inverse" of an element x in a monoid X = <X, *, e> is an element y C X such that x*y = e = y*x.  An element that has an inverse in X is said to be "invertible" (relative to * and e).  If x C X has an inverse, then it is unique to x.  To see this, suppose that y' is also an inverse of x.  Then it follows that: